Question

Find the greatest number which divides 764 and 1198, leaving remainders 8 and 10 respectively.

Hint:

Using Euclid's division algorithm is a fast way to verify HCF for large integers:
\[ 1188 = 756 \times 1 + 432 \] \[ 756 = 432 \times 1 + 324 \] \[ 432 = 324 \times 1 + 108 \] \[ 324 = 108 \times 3 + 0 \] Since the last non-zero remainder is 108, this confirms that the HCF is indeed 108!

Answer 1

Step 1: Reduce to an HCF problem.
The required number divides $764-8=756$ and $1198-10=1188$ exactly, so it is the HCF of $756$ and $1188$.
Step 2: Apply Euclid's division algorithm.
$1188=756\times1+432$, then $756=432\times1+324$.
Step 3: Keep dividing until the remainder is zero.
$432=324\times1+108$, then $324=108\times3+0$.
Step 4: Read off the HCF.
The last non-zero remainder is $108$, so the greatest number required is $108$.
\[ \boxed{108} \]